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On metric Ramsey-type phenomena
The main question studied in this article may be viewed as a nonlinear
analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey
theory in combinatorics. Given a finite metric space on n points, we seek its
subspace of largest cardinality which can be embedded with a given distortion
in Hilbert space. We provide nearly tight upper and lower bounds on the
cardinality of this subspace in terms of n and the desired distortion. Our main
theorem states that for any epsilon>0, every n point metric space contains a
subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space
with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the
distortion is tight up to the log(1/\epsilon) factor. We further include a
comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio
Recommended from our members
Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their
properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic
and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization,
Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions.
This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
Recommended from our members
Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combinatorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
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Combinatorics, Probability and Computing
One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b
Dissipative Kerr solitons in optical microresonators
This chapter describes the discovery and stable generation of temporal
dissipative Kerr solitons in continuous-wave (CW) laser driven optical
microresonators. The experimental signatures as well as the temporal and
spectral characteristics of this class of bright solitons are discussed.
Moreover, analytical and numerical descriptions are presented that do not only
reproduce qualitative features but can also be used to accurately model and
predict the characteristics of experimental systems. Particular emphasis lies
on temporal dissipative Kerr solitons with regard to optical frequency comb
generation where they are of particular importance. Here, one example is
spectral broadening and self-referencing enabled by the ultra-short pulsed
nature of the solitons. Another example is dissipative Kerr soliton formation
in integrated on-chip microresonators where the emission of a dispersive wave
allows for the direct generation of unprecedentedly broadband and coherent
soliton spectra with smooth spectral envelope.Comment: To appear in "Nonlinear optical cavity dynamics", ed. Ph. Grel
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